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A188431
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The number of n-full sets, F(n).
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45
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1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 1, 2, 1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 10, 13, 14, 17, 20, 25, 28, 34, 40, 46, 54, 62, 69, 80, 90, 102, 115, 131, 144, 167, 186, 213, 239, 273, 304, 349, 388, 441, 495, 563, 625, 710, 790, 890, 990, 1114, 1232, 1387, 1530, 1713, 1894, 2119, 2330, 2605, 2866, 3192, 3512, 3910, 4289, 4774, 5237, 5809, 6377, 7068, 7739
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OFFSET
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0,13
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COMMENTS
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Let A be a set of positive integers. We say that A is n-full if (sum A)=[n] for a positive integer n, where (sum A) is the set of all positive integers which are a sum of distinct elements of A and [n]={1,2,...,n}. Then F(n) denotes the number of n-full sets.
Also the number of distinct and complete partitions of n, by definition, which are counted by A000009 and A126796. - George Beck, Nov 06 2017
An integer partition of n is complete (see also A325781) if every number from 0 to n is the sum of some submultiset of the parts. The Heinz numbers of these partitions are given by A325986. - Gus Wiseman, May 31 2019
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LINKS
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L. Naranjani and M. Mirzavaziri, Full Subsets of N, Journal of Integer Sequences, 14 (2011), Article 11.5.3.
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FORMULA
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F(n) = Sum_(i=L(n) .. U(n), F(n,i)), where F(n,i) = Sum_(j=L(n-i) .. min(U(n-i),i-1), F(n-i,j) ) and L(n), U(n) are defined in A188429 and A188430, respectively.
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1+x^k), with a(0)=1. - Paul D. Hanna, Mar 08 2012
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(3/4), where c = 0.03316508... - Vaclav Kotesovec, Oct 21 2019
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EXAMPLE
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a(26) = 10, because there are 10 26-full sets: {1,2,4,5,6,8}, {1,2,3,5,7,8}, {1,2,3,5,6,9}, {1,2,3,4,7,9}, {1,2,3,4,6,10}, {1,2,3,4,5,11}, {1,2,4,8,11}, {1,2,4,7,12}, {1,2,4,6,13}, {1,2,3,7,13}.
G.f.: 1 = 1/(1+x) + 1*x/((1+x)*(1+x^2)) + 0*x^2/((1+x)*(1+x^2)*(1+x^3)) + 1*x^3/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) +...+ a(n)*x^n / Product_{k=1..n+1} (1+x^k) +...
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MAPLE
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sums:= proc(s) local i, m;
m:= max(s[]);
`if`(m<1, {}, {m, seq([i, i+m][], i=sums(s minus {m}))})
end:
a:= proc(n) local b;
b:= proc(i, s) local si;
if i=1 then `if`(sums(s)={$1..n}, 1, 0)
else si:= s union {i};
b(i-1, s)+ `if`(max(sums(si)[])>n, 0, b(i-1, si))
fi
end; b(n, {1})
end:
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,
b(n, i-1)+`if`(i>n or i>n-i+1, 0, b(n-i, i-1))))
end:
a:= n-> b(n$2):
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MATHEMATICA
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Sums[s_] := Sums[s] = With[{m = Max[s]}, If[m < 1, {}, Union @ Flatten @ Join[{m}, Table[{i, i + m}, {i, Sums[s ~Complement~ {m}]}]]]];
a[n_] := Module[{b}, b[i_, s_] := b[i, s] = Module[{si}, If[i == 1, If[Sums[s] == Range[n], 1, 0], si = s ~Union~ {i}; b[i-1, s] + If[Max[ Sums[si]] > n, 0, b[i - 1, si]]]]; b[n, {1}]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Union[Total/@Union[Subsets[#]]]==Range[0, n]&]], {n, 30}] (* Gus Wiseman, May 31 2019 *)
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PROG
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(PARI) /* As coefficients in g.f. */
{a(n)=local(A=[1]); for(i=1, n+1, A=concat(A, 0); A[#A]=polcoeff(1 - sum(m=1, #A, A[m]*x^m/prod(k=1, m, 1+x^k +x*O(x^#A) )), #A) ); A[n+1]}
for(n=0, 50, print1(a(n), ", ")) /* Paul D. Hanna, Mar 06 2012 */
(Haskell)
import Data.MemoCombinators (memo2, integral, Memo)
a188431 n = a188431_list !! (n-1)
a188431_list = map
(\x -> sum [fMemo x i | i <- [a188429 x .. a188430 x]]) [1..] where
fMemo = memo2 integral integral f
f _ 1 = 1
f m i = sum [fMemo (m - i) j |
j <- [a188429 (m - i) .. min (a188430 (m - i)) (i - 1)]]
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CROSSREFS
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Cf. A002033, A103295, A108917, A276024, A325763, A325765, A325781, A325782, A325788, A325986, A325790, A325791.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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